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Equivariant Nielsen Invariantsfor Discrete Groups Pacific Journal of Mathematics EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS JULIA WEBER Volume 231 No. 1 May 2007 PACIFIC JOURNAL OF MATHEMATICS Vol. 231, No. 1, 2007 EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS JULIA WEBER For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f : X → X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic to a fixed point free G-map if the gener- alized equivariant Lefschetz invariant λG( f ) is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem. 1. Introduction The Lefschetz number is a classical invariant in algebraic topology. If the Lefschetz number L( f ) of an endomorphism f : X → X of a compact CW-complex is nonzero, then f has a fixed point. This is the famous Lefschetz fixed point theorem. The converse does not hold: If the Lefschetz number of f is zero, we cannot conclude f to be fixed point free. A more refined invariant which allows to state the converse is the Nielsen num- ber: The Nielsen number N( f ) is zero if and only if f is homotopic to a fixed point free map. More generally, the Nielsen number is used to give precise minimal bounds for the number of fixed points of maps homotopic to f . Its development was started by Nielsen [1920]; a comprehensive treatment can be found in [Jiang 1983]. We are interested in the equivariant generalization of these results. Given a discrete group G and a G-equivariant endomorphism f : X → X of a finite proper G-CW-complex, we introduce equivariant Nielsen invariants called NG( f ) and N G( f ). They are equivariant analogs of the Nielsen number and are derived from the generalized equivariant Lefschetz invariant λG( f ) [Weber 2005, Defi- nition 5.13]. MSC2000: primary 55M20, 57R91; secondary 54H25, 57S99. Keywords: Nielsen number, discrete groups, equivariant, Lefschetz fixed point theorem. 239 240 JULIA WEBER We proceed to show that these Nielsen invariants give minimal bounds for the number of orbits of fixed points in the G-homotopy class of f . One even obtains results concerning the type and “location” (connected component of the relevant fixed point set) of these fixed point orbits. These lower bounds are sharp if X is a cocompact proper smooth G-manifold satisfying the standard gap hypotheses. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem: If X is a G-Jiang space as defined in Definition 5.2, then LG( f ) = 0 implies that f is G-homotopic to a fixed point free map. Here, LG( f ) is the equivariant Lefschetz class [Luck¨ and Rosenberg 2003a, Definition 3.6], the equivariant analog of the Lefschetz number. These results were motivated by work of Luck¨ and Rosenberg [2003a; 2003b]. For G a discrete group and an endomorphism f of a cocompact proper smooth G-manifold M, they prove an equivariant Lefschetz fixed point theorem [2003a, Theorem 0.2]. The converse of that theorem is proven here. Another motivation for the present article is the fact that the algebraic approach to the equivariant Reidemeister trace provides a good framework for computation. The connection to the machinery used in the study of transformation groups [Luck¨ 1989; tom Dieck 1987] allows results to translate more readily from transformation groups to geometric equivariant topology and vice-versa. When G is a compact Lie group, Wong [1993] obtains results on equivariant Nielsen numbers which strongly influenced us. The main difference between our work and Wong’s is that we treat possibly infinite discrete groups. Another dif- ference is that our approach is more structural. We can read off the equivariant Nielsen invariants from the generalized equivariant Lefschetz invariant λG( f ). In case G is a finite group, Ferrario [2003] studies a collection of generalized Lefschetz numbers which can be thought of as an equivariant generalized Lefschetz number. In contrast to the generalized equivariant Lefschetz invariant λG( f ) these do not incorporate the WH-action on the fixed point set X H . A generalized Lef- schetz trace for equivariant maps has also been defined by Wong, as mentioned in [Hart 1999]. For compact Lie groups, earlier definitions of generalized Lefschetz numbers for equivariant maps were made in [Wilczynski´ 1984; Fadell and Wong 1988]. These authors used the collection of generalized Lefschetz numbers of the maps f H , for H < G. In general, these numbers are not sufficient since they do not take the equivariance into account adequately. For further reading on equivariant fixed point theory, see [Ferrario 2005], where an extensive list of references is given. Organization of paper. In Section 2, we introduce the generalized equivariant Lef- schetz invariant. We briefly assemble the concepts and definitions which are needed for the definition of equivariant Nielsen invariants in Section 3. EQUIVARIANT NIELSEN INVARIANTS FOR DISCRETE GROUPS 241 The equivariant Nielsen invariants give lower bounds for the number of fixed point orbits of f . This is shown in Section 4. The standard gap hypotheses are introduced, and it is shown that under these hypotheses these lower bounds are sharp. In the nonequivariant case, we know that the generalized Lefschetz invariant is the right element when looking for a precise count of fixed points. We read off the Nielsen number from this invariant. In general, the Lefschetz number contains too little information. But under certain conditions, we can conclude facts about the Nielsen number from the Lefschetz number directly. These are called the Jiang conditions [1983, Definition II.4.1] (see also [Brown 1971, Chapter VII]). In Section 5, we introduce the equivariant version of these conditions. We also give examples of G-Jiang spaces. In Section 6, we derive equivariant analogs of statements about Nielsen numbers found in [Jiang 1983], generalizing results from [Wong 1993] to infinite discrete groups. In particular, if X is a G-Jiang space, the converse of the equivariant Lefschetz fixed point theorem holds. 2. The generalized equivariant Lefschetz invariant Classically, the Nielsen number is defined geometrically by counting essential fixed point classes [Brown 1971, Chapter VI; Jiang 1983, Definition I.4.1]. Alternatively one defines it using the generalized Lefschetz invariant. Let X be a finite CW-complex, let f : X → X be an endomorphism, let x be a basepoint of X, and let X λ( f ) = nα · α ∈ ޚπ1(X, x)φ α∈π1(X,x)φ be the generalized Lefschetz invariant associated to f [Reidemeister 1936; Wecken 1941], where −1 ޚπ1(X, x)φ := ޚπ1(X, x)/φ(γ )αγ ∼ α, with γ, α ∈ π1(X, x). Here φ is the map induced by f on the fundamental group π1(X, x). We have φ(γ ) = wf (γ )w−1, where w is a path from x to f (x). The generalized Lefschetz invariant is also called Reidemeister trace in the literature, which goes back to the original name “Reidemeistersche Spureninvariante” used by Wecken. The set π1(X, x)φ is often denoted by ᏾( f ) and called set of Reidemeister classes of f . Since we will introduce a variation of this set in Definition 2.3, we prefer to stick with the notation used in [Weber 2005]. 242 JULIA WEBER Definition 2.1. The Nielsen number of f is defined by N( f ) := # α nα 6= 0 . The Nielsen number is the number of classes in π1(X, x)φ with nonzero coeffi- cients. A class α with nonzero coefficient corresponds to an essential fixed point class in the geometric sense. In the equivariant setting, the fundamental category replaces the fundamental group. The fundamental category of a topological space X with an action of a discrete group G is defined as follows [Luck¨ 1989, Definition 8.15]. Definition 2.2. Let G be a discrete group, and let X be a G-space. Then the fundamental category 5(G, X) is the following category: • The objects Ob5(G, X) are G-maps x : G/H → X, where the H ≤ G are subgroups. • The morphisms Morx(H), y(K ) are pairs (σ, [w]), where – σ is a G-map σ : G/H → G/K – [w] is a homotopy class of G-maps w : G/H × I → X relative G/H ×∂ I such that w1 = x and w0 = y ◦ σ . The fundamental category is a combination of the orbit category of G and the fundamental groupoid of X. If X is a point, then the fundamental category is just the orbit category of G, whereas when G is the trivial group, the definition reduces to the definition of the fundamental groupoid of X. We often view x as the point x(1H) in the fixed point set X H . We call X H (x) the connected component of X H containing x(1H). We also consider the relative H >H >H H fixed point set, the pair X (x), X (x) . Here X (x) = {z ∈ X (x) | Gz 6= H} is the singular set, where Gz denotes the isotropy group of z. In order to H simplify notation, we use f (x) to denote f |X H (x), and we use fH (x) instead of f | . X H (x),X >H (x) Fixed points of f can only exist in X H (x) when X H ( f (x)) = X H (x), i.e, when the points f (x) and x lie in the same connected component of X H . Definition 2.3.
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